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Spectral identities for Schrödinger operators

Namig J. Guliyev

TL;DR

This work addresses inverse spectral problems for a one-dimensional Schrödinger operator on (0,π) with boundary conditions that depend on the spectral parameter through rational Herglotz–Nevanlinna functions. It derives a finite system of algebraic identities that connect boundary data to spectral data (eigenvalues and norming constants) via moments σ_k and kernel coefficients ω, and proves that the system is uniquely solvable, enabling direct reconstruction of the boundary data from spectral information. Interpreted as a boundary-version of the Gelfand–Levitan framework, the identities provide a compact route to boundary coefficient recovery and extend to incomplete data scenarios. The results complement endpoint symmetry, recover known special cases (e.g., ind f = 0), and have potential applications in inverse problems and isospectral theory.

Abstract

We obtain a system of identities relating boundary coefficients and spectral data for the one-dimensional Schrödinger equation with boundary conditions containing rational Herglotz--Nevanlinna functions of the eigenvalue parameter. These identities can be thought of as a kind of mini version of the Gelfand--Levitan integral equation for boundary coefficients only.

Spectral identities for Schrödinger operators

TL;DR

This work addresses inverse spectral problems for a one-dimensional Schrödinger operator on (0,π) with boundary conditions that depend on the spectral parameter through rational Herglotz–Nevanlinna functions. It derives a finite system of algebraic identities that connect boundary data to spectral data (eigenvalues and norming constants) via moments σ_k and kernel coefficients ω, and proves that the system is uniquely solvable, enabling direct reconstruction of the boundary data from spectral information. Interpreted as a boundary-version of the Gelfand–Levitan framework, the identities provide a compact route to boundary coefficient recovery and extend to incomplete data scenarios. The results complement endpoint symmetry, recover known special cases (e.g., ind f = 0), and have potential applications in inverse problems and isospectral theory.

Abstract

We obtain a system of identities relating boundary coefficients and spectral data for the one-dimensional Schrödinger equation with boundary conditions containing rational Herglotz--Nevanlinna functions of the eigenvalue parameter. These identities can be thought of as a kind of mini version of the Gelfand--Levitan integral equation for boundary coefficients only.

Paper Structure

This paper contains 2 sections, 1 theorem, 31 equations.

Table of Contents

  1. Introduction and main result
  2. Proof

Key Result

Theorem 1

The following identities hold: with the convention that $\omega_0 = 1$ and $\omega_m = 0$ if $m < 0$. Moreover, this system is uniquely solvable, both when either $\omega_1$, $\ldots$, $\omega_{\mathop{\mathrm{ind}}\nolimits f + 1}$ or $\sigma_0$, $\ldots$, $\sigma_{\mathop{\mathrm{ind}}\nolimits f}$ are treated as variables.

Theorems & Definitions (3)

  • Theorem
  • Remark 1
  • Remark 2